nLab Eilenberg-MacLane space type

Contents

Context

Type theory

natural deduction metalanguage, practical foundations

  1. type formation rule
  2. term introduction rule
  3. term elimination rule
  4. computation rule

type theory (dependent, intensional, observational type theory, homotopy type theory)

syntax object language

computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory

logiccategory theorytype theory
trueterminal object/(-2)-truncated objecth-level 0-type/unit type
falseinitial objectempty type
proposition(-1)-truncated objecth-proposition, mere proposition
proofgeneralized elementprogram
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
cut elimination for implicationcounit for hom-tensor adjunctionbeta reduction
introduction rule for implicationunit for hom-tensor adjunctioneta conversion
logical conjunctionproductproduct type
disjunctioncoproduct ((-1)-truncation of)sum type (bracket type of)
implicationinternal homfunction type
negationinternal hom into initial objectfunction type into empty type
universal quantificationdependent productdependent product type
existential quantificationdependent sum ((-1)-truncation of)dependent sum type (bracket type of)
equivalencepath space objectidentity type/path type
equivalence classquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
-0-truncated higher colimitquotient inductive type
coinductionlimitcoinductive type
presettype without identity types
completely presented setdiscrete object/0-truncated objecth-level 2-type/set/h-set
setinternal 0-groupoidBishop set/setoid
universeobject classifiertype of types
modalityclosure operator, (idempotent) monadmodal type theory, monad (in computer science)
linear logic(symmetric, closed) monoidal categorylinear type theory/quantum computation
proof netstring diagramquantum circuit
(absence of) contraction rule(absence of) diagonalno-cloning theorem
synthetic mathematicsdomain specific embedded programming language

homotopy levels

semantics

Induction

Contents

Idea

Let GG be an abelian group, and nn a natural number. The Eilenberg-MacLane space K(G,n)K(G,n) is the unique space that has π n(K(G,n))G\pi_n(K(G,n))\cong G and its other homotopy groups trivial. These spaces are a basic tool in classical algebraic topology, they can be used to define cohomology.

There is an analogous construction in homotopy type theory.

Definition

Let GG be a group. The Eilenberg-MacLane space type K(G,1)K(G,1) is the higher inductive 1-type with the following constructors:

  • base:K(G,1)base : K(G,1)
  • loop:G(base=base)loop : G \to (base = base)
  • id:loop(e)=refl baseid : loop(e) = refl_{base}
  • comp: (x,y:G)loop(xy)=loop(y)loop(x)comp : \prod_{(x,y:G)} loop(x \cdot y) = loop(y) \circ loop(x)

basebase is a point of K(G,1)K(G,1), looploop is a function that constructs a path from basebase to basebase for each element of GG. idid says the path constructed from the identity element is the trivial loop. Finally, compcomp says that the path constructed from group multiplication of elements is the concatenation of paths.

It should be noted that this type is 1-truncated which could be added as another constructor:

  • x,y:K(G,1) p,q:x=y r,s:p=qr=s\displaystyle \prod_{x,y:K(G,1)} \prod_{p,q:x=y} \prod_{r,s:p=q} r = s

Recursion principle

To define a function f:K(G,1)Cf : K(G,1) \to C for some type CC, it suffices to give

  • a point c:Cc : C
  • a family of loops l:G(c=c)l : G \to (c = c)
  • a path l(e)=idl(e)=id
  • a path l(xy)=l(y)l(x)l(x \cdot y) = l(y) \circ l(x)
  • a proof that CC is a 1-type.

Then ff satisfies the equations

f(base)cf(base) \equiv c
ap f(loop(x))=l(x)ap_{f} (loop(x))=l(x)

It should be noted that the above can be compressed, to specify a function f:K(G,1)Cf : K(G,1) \to C, it suffices to give:

  • a proof that CC is a 1-type
  • a point c:Cc : C
  • a group homomorphism from GG to Ω(C,c)\Omega(C,c)

See also

References

Formalization that K(G,n)K(G,n) is a spectrum here

Last revised on June 9, 2022 at 02:13:31. See the history of this page for a list of all contributions to it.